A Reconstruction theorem for genus zero Gromov-Witten invariants of stacks

We generalize the {\it First Reconstruction Theorem} of Kontsevich and Manin in two respects. First, we allow the target space to be a Deligne-Mumford stack. Second, under some convergence assumptions, we show it suffices to check the hypothesis of $H^2$-generation not on the cohomology ring, but on an any quantum ring in the family given by small quantum cohomology. As an example the latter result is used to compute genus zero Gromov-Witten invariants of ${\Bbb P}(1,b)$.